Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.
h(x)=e^x(x+1)^3

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value. This typically happens when the function is undefined at that point, often due to division by zero. Identifying vertical asymptotes involves finding values of x that make the denominator of a rational function zero, while ensuring the numerator is not also zero at those points.

Exponential Functions

Exponential functions, such as h(x) = e^x, are characterized by a constant base raised to a variable exponent. These functions grow rapidly and are always positive, meaning they do not have vertical asymptotes. Understanding their behavior is crucial when analyzing composite functions, as they can influence the overall shape and limits of the function.

Polynomial Functions

Polynomial functions, like (x+1)^3, are expressions consisting of variables raised to whole number powers. They are continuous and defined for all real numbers, meaning they do not have vertical asymptotes. Analyzing the degree and roots of polynomial functions helps in understanding their behavior, particularly in conjunction with other function types in composite expressions.